Photons emitted when He^+ ions make the transition from the n=3 to the n=1 level fall on a metal plate of work function 4.0 eV. The fastest…

For a hydrogen-like ion $E_n=-13.6\,Z^2/n^2\ \mathrm{eV}$, so for $\mathrm{He^+}$ ($Z=2$) the $3\to1$ photon has energy $13.6\times4\times\left(1-\tfrac19\right)=48.36\ \mathrm{eV}$. The maximum kinetic energy of photoelectrons is $K=48.36-4.0=44.36\ \mathrm{eV}$ (Einstein's equation). In the field the magnetic force supplies the centripetal force, $r=\dfrac{mv}{eB}=\dfrac{\sqrt{2mK}}{eB}$. With $K=44.36\times1.6\times10^{-19}\ \mathrm{J}$ this gives $\sqrt{2\times9.11\times10^{-31}\times7.10\times10^{-18}}\approx3.60\times10^{-24}\ \mathrm{kg\,m\,s^{-1}}$ and $r\approx\dfrac{3.60\times10^{-24}}{1.6\times10^{-19}\times10^{-3}}\approx2.25\ \mathrm{cm}$. Using the photon energy instead of $K$ gives $2.35\ \mathrm{cm}$, treating the ion as hydrogen ($Z=1$, $K=8.09\ \mathrm{eV}$) gives $0.96\ \mathrm{cm}$, and halving $K$ gives $1.59\ \mathrm{cm}$. The speed of the electrons is about $4\times10^{6}\ \mathrm{m\,s^{-1}}$, so the non-relativistic formula is adequate.